**3. Conclusions**

We presented a new microring resonator based on only one multimode waveguide. The critical control of the microring resonator can be achieved by using graphene-silicon waveguide. The proposed device has all characteristics of a traditional microring resonator. Some important parameters of the proposed device such as finesse, quality factor, etc. are also presented in this chapter. The device operation has been verified by using the FDTD. This microring resonator structure is very compact and can be useful for further applications in optical switching, filtering, and sensing.

### **Acknowledgement**

where Δ*λFWHM* is the resonance full width at half maximum and FSR is the free spectral range. The free spectral range is the distance between two peaks on a wavelength scale. By differentiating the equation *φ* ¼ *βLR*, we get

*Electromagnetic Propagation and Waveguides in Photonics and Microwave Engineering*

**Figure 9** shows the finesse and quality factor with different chemical potential at a radius of 5*μm*. We see that a maximum finesse and quality factor can be achieved

The normalized transmissions of the propose microring resonator in **Figure 1** at microring radii of 5*μm* and 50*μm* are shown in **Figure 10**. Here we assume that the chemical potential is *μ<sup>c</sup>* ¼ 0*:*45*eV*. The simulations show that the exact characteris-

Finally, we use FDTD method to simulate the proposed microring resonator based on multimode waveguide. In our FDTD simulations, we take into account the wavelength dispersion of the silicon waveguide. A light pulse of 15 fs pulse width is launched from the input to investigate the transmission characteristics of the device. The grid sizes Δx = Δy = Δz = 20 nm are chosen in our simulations for

*Optical field propagation through the coupler for input signal presented at (a) port 1 and (b) port 2.*

*<sup>d</sup><sup>λ</sup>* .

*ngL*, where the group index *ng* <sup>¼</sup> *neff* � *<sup>λ</sup> dneff*

*Transmissions of the microring resonator with two microring radii of* 5*μm and* 50*μm.*

tics of a single microring resonator can be achieved.

at a chemical potential of *μ<sup>c</sup>* ¼ 0*:*57*eV*.

*FSR* <sup>¼</sup> *<sup>λ</sup>*<sup>2</sup>

**Figure 11.**

**58**

**Figure 10.**

This research is funded by the Ministry of Natural Resources and Environment of Vietnam under the project BĐKH.30/16-20.
